Local trace formulae and scaling asymptotics for general quantized Hamiltonian flows

A class of trace formulae in the Toeplitz quantization of Hamiltonian flows on compact symplectic manifolds is given a local intepretation in terms of certain scaling asymptotics near a symplectic fixed locus; these are reminiscent of the near diagonal asymptotics of equivariant Szegö kernels. In spectral theory, the distributional trace of a positive elliptic operator encapsulates information on the asymptotic distribution of its eigenvalues; a trace formula relates the singularities of the trace to Poincaré data of an appropriate Hamiltonian flow. This leads to the asymptotics of certain smoothing kernels, related to the Fourier transform of the trace of a wave operator. In Toeplitz quantization, we consider analogues of these, with the wave operator replaced by a quantized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the concentration of these smoothing kernels near the fixed loci of the linearized dynamics, and specifically how the Poincaré map controls their scaling asymptotics. This generalizes previous results on holomorphic flows.